Meaning
A Hamiltonian-bearing admissible comparison transports the energy structure as well as the carrier and observable algebra.
It retains the exact-ground projection; electric, magnetic, and seam descent; the Hamiltonian cells comparing the two routes around mixed squares; higher composition data; and all three residual budgets. If a budget changes, that change must appear in the adjusted coercive estimates.
Example (illustrative)
Two routes may identify the same carrier by the same unitary while their Hamiltonian restrictions differ by a seam residual. The comparison must record and control the residual; carrier agreement alone does not settle the energy comparison.
Scope and limits
Carrier coherence and Hamiltonian coherence are separately typed requirements. Neither can stand in for the other merely because some representing matrices coincide.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 2.1.